Download Grade 7/8 Math Circles Angles Angles

Faculty of Mathematics
Waterloo, Ontario N2L 3G1
Centre for Education in
Mathematics and Computing
Grade 7/8 Math Circles
October 14/15, 2014
Angles
Angles
An angle is the space between two lines that intersect each other.
Terms and angles you should know:
An acute angle is an angle less then 90°.
A right angle is a 90°angle.
An obtuse angle is more than 90°but less then 180°
A straight angle is a 180°angle.
Classify these angles as on of the above:
1
a)
b)
c)
d)
e)
Two angles are complementary if they add up to 90°. Below, a and b are complementary
since a + b = 90
Find the complementary angle for the given angles(find a):
a)
b)
c)
d)
Two angles are supplementaryif they add up to 180°. Below, a and b are supplementary
since a + b = 180
Find the supplementary angle for the given angles(find a):
2
a)
b)
c)
d)
Trick to remember: C comes before S and 90 comes before 180
An opposite angle is the angle that is directly opposite from a given angle when two lines
intersect.
Can we explain this using our knowledge of supplementary angles?
A transversal is a line that intersects two or more lines. If these lines are parallel, the
angles around these lines have very nice properties.
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Corresponding angles are angles that are on the same side of the traversal as well as on the
same side of each parallel line. Corresponding angles are equal to each other. We sometimes
refer to these as F angles.
In the picture above there are 2 pairs of corresponding angles: D=H and B=F.
Alternate angles are angles that are on opposite sides of the transversal and on opposite
sides of the parallel lines. Alternate angles are equal to each other. We sometimes refer to
these as Z angles.
In the picture above there are 2 pairs of corresponding angles: C=F and A=H.
Interior angles are angles on the same side of a transversal and on opposite sides of the
parallel lines. These angles add up to 180°. These are sometimes referred to as C angles.
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In the picture above D and F are interior angles. D + F = 180
Find the missing angles.
Triangles and More
First, lets talk about the 3 types of triangles.
Equilateral: These triangles have three equal length sides and three equal angles.
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Isosceles: These triangles have two equal length sides and two equal angles.
Scalene: No sides are equal and no angles are equal.
What is the sum of the interior angles of a triangle? 180. Any triangle, regardless of size
has 3 interior angles that add up to 180.
Using our knowledge of angles (interior, alternate, supplementary . . . ) can we show why this
is true? Why is is that with any given triangle the angles always add up to 180.
Lets try extending the lines and look at this again.
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Now can you show the angles add up to 180?
If we consider the angle x + y we can say this angle is supplementary to z(we also could of
said x + z is supplementary to y. This means that x + y + z. Notice, we did this is unknown
angles, this means that it works for any x, y and z.
What is the sum of angles in a parallelogram? Can you show this using triangles? The sum
of the angles in a parallelogram is equal to 360 and we can show this using triangles. Draw
a square, and from one corner draw a line to the opposite corner.
Each triangle has 180 degrees of interior angles, and there are two of them so 180+180 = 360.
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What about a pentagon? Start with a pentagon, chose a corner and draw a line to each
corner that is not directly beside it.
Using this same idea, we see that the sum of the angles inside a pentagon is
180 + 180 + 180 = 540. Can you think of how this might extend further? What would the
sum of interior angles be in a heptagon(7 sides), what about a decagon(10 sides)?
Radians
Until now, all the angle we have seen were in degrees. However, angles like many other measurements can have different unit (similar to kilometers and miles). For angles, a commonly
used unit is the radian. One radian is the angle formed when the arc length on a circle is
equal to the radius. How many radians in one full circle?
Looking at the arc length formula
n
× 2πr here n is angle made by the arc length.
a = 360
In this formula we have the number 360. This is the number of degrees in a circle. To
show how many radians we have in a circle lets replace that with a variable rad. Giving us
n
a = rad
× 2πr
Now n is the number of degrees, or in our case radians, and we defined 1 radian as then
angle when the arc length is the same as the radius. In other words when a = r. For ease of
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calculation lets make that be 1, a = r = 1. Our formula is now:
1
1 = rad
× 2π × 1
rad = 2π
Multiply both sides by rad
Therefore there are 2π radians in a circle. This lead the equality 2π = 360°
Conversion
We can use this equality to convert from degrees to radians and back. For example, convert
60°to radians. If we take the formula and divide by 360 on each side we get
2π
=1
360
2π×60
=
360
2π
= 60
6
π
= 60
3
60
Then we multiply each side by the desired angle
simplify
So 60 degrees is equal to
π
3
radians.
Going from radians to degrees we can do something similar. For example, convert
degrees. We start by dividing each side by 2 to get
π = 180
Now we want π4 so we multiply each side by 14
π
= 180
and simplify
4
4
π
= 45
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Examples Convert the following to either degrees or radians
1. π
2. 120°
4π
3.
3
4. 300◦
5.
12π
6
6.
π
18
7. 20°
9
π
4
to
PROBLEMS
1. Identify the type of angles.
a) Obtuse
b) Right
c) Obtuse
d) Acute
e) Straight
2. What is a complementary angle? What is a supplementary angle? Two angles are
complementary if they add up to 90°. Two angles are supplementary if they add up to
180°
3. (a) What is the complementary angle of 30°? 60
(b) What is the complementary angle of 17°? 73
(c) What is the supplementary angle of 25°? 155
(d) What is the supplementary angle of 111°? 69
(e) What is the supplementary angle of 90°? 90
(f) If angle w and x are complementary and y and z are complementary what are x
and (w + y + z)? They are supplementary
(g) If x is supplementary to 42°and y is complementary to 42°what is x + y? 186
4. List all pairs of angles that are supplementary to each other.
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(A,B), (A,C), (A,G), (A,F),
(B,D), (B,H), (B,E),
(C,D), (C,E), (C,H) ,
(D,F), (D,G)
(E,F), (E,G),
(F,H),
(G,H)
5. Find the missing angles.
a) x = 60 , y = 120
b) x = 75 , y = 75
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c) x = 30
d) x = 120
6. Find the missing angles.
a) x = 25
b) x = 67.5
c) x = 60
d) x = 110, y = 22.5
7. Find x and y in the following hexagon:
x = 30, y = 60
8. Two angles are supplementary to each other. One is 56 degrees larger then the other.
What is the smallest angle.
62
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9. You order a pizza and it is cut into 10 even pieces. If you were to measure the angle
of a slice, what would it be in radians and in degrees?
π
36°or rad
5
10. Convert
(a) 3π
(b) 190°
540°
19π
18
4π
(c)
6
120°
(d) 270◦
3π
2
(e)
8π
9
160°
(f)
π
18
10°
(g) 20°
π
18
11. Find a pattern in the sum of angles inside a triangle, square, pentagon . . . Using this
pattern What is the sum of interior angles in a 92 faced figure?
The pattern is (n − 2) × 180 Therefore in a 92 faced figure, the sum of interior angles
is:
= (n − 2) × 180
= (92 − 2) × 180
= (90) × 180
= 16200
12. ** The ∠A is three times the size ∠B and ∠B is twice the size of ∠C. If ∠A and two
times ∠C are supplementary what is ∠B?
Here we have A = 3 × B and B = 2 × C. Knowing A and C are supplementary we
can write A + 2C = 180. Replace A and C and we get 3B + B = 180. This implies
that B = 45
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